An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons - Wang, Haitao (Discrete & computational geometry, 2023)
Given in the plane a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for S in P . It is known that the problem has an Ω ( n + m log m ) time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that solves the problem in O ( n + m log m ) expected time. The previous best deterministic algorithms solve the problem in O ( n log log n + m log m ) time [Oh, Barba, and Ahn, SoCG 2016] or in O ( n + m log m + m log 2 n ) time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm that takes O ( n + m log m ) time, which is optimal. This answers affirmatively an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.